Impulse Synthetic Aperture Radar
4.2 Timed Array Analysis
Let us consider an array, composed of 2N+1 elements, as depicted in Figure 4�1� Each element is excited by the same signal f t( )� In the far field, the radiated field is propor-tional to
F t f t n t
n N
N
( , )ϑ = ( − )
∑
=− ∆ (4�1)where ∆t a= sin ϑ , and c is the speed of light in vacuum�c
Impulse Synthetic Aperture Radar 87
A definition of the radiation diagram similar to that of sinusoidal field is possible by substituting its radiated power density with energy density as follows (Franceschetti, Tatoian, and Gibbs 2005):
g F t t
F t t
( ) ( , )d ( , )d
ϑ ϑ
=
∫
∫
2
2 0 (4�2)
As an illustrative example, the case N = 2 is depicted in Figure 4�2, where the feeding signal f t( ) is the rectangular pulse, f t( )= rect[t T], T being the pulsewidth�
Furthermore, in the far field, that is, in the properly defined Fraunhofer region of the array in time domain, r> (2L2 2cT), we have 2N t∆ =(2Na c)sinθ≅(L c)sinθ≤T, L=(2N+1)a≅2Na being the array length� These equalities are valid for large arrays, which is the case hereafter�
Computation of the radiation diagram, Equation 4�2, is now in order� From Equation 4�1, it follows that
F t2( , )d0 t (2N 1)2T
∫
= + (4�3)n = −N n = −1 n = 1 n = N
La θ
FIgure 4.1
Array composed of 2N+1 identical elements� (From Franceschetti, G�, J� Tatoian, and G� Gibbs, Timed arrays in a nutshell, IEEE Trans Antennas Propagat 53(12):4073–82), © (2005) IEEE�)
F(t,0)
(a)
(b)
(c) n = 0
F(t,θ)
T f (t)
−2−1
12
Δt
t tt t tt
t
FIgure 4.2
Timed array excited by a rectangular pulse and the resulting far-field radiated signals: (a) Received and super-posed pulses along the broadside direction; (b) received pulses along the direction θ; (c) supersuper-posed received pulses along the direction θ� (From Franceschetti, G�, J� Tatoian, and G� Gibbs, Timed arrays in a nutshell, IEEE Trans Antennas Propagat 53(12):4073–82), © (2005) IEEE�)
88 Advances in Environmental Remote Sensing
where the last equality is valid for large arrays� Computation of Equation 4�4 makes refer-ence to the diagrams depicted in Figure 4�2� The integral is broken down into two parts:
the first is the time interval where the pulses are synchronized; the second part is relative to the remaining time-interval, and is obtained by the evaluation of a finite summation (Franceschetti, Tatoian, and Gibbs 2005)�
Dividing Equation 4�4 by Equation 4�3 leads to the formal expression of the radiation diagram; see Equation 4�2, hence
Letting g( )Θ = 1 2 and solving for the (conventional) 3-dB beamwidth 2Θ, we obtain
sinΘ Θ≅ =3 , Θ=
2cT 2 3
L
cT
L (4�6)
The definition in Equation 4�2 for the radiation diagram emphasizes the power content of the radiated beam� The alternative definition
refers to the shape of the pulse, ξ being the similarity factor in a quadratic norm (Franceschetti, Tatoian, and Gibbs 2005)� Assuming as convenient value for the similarity factor ξ(Θ) = 0�917, and solving for the beamwidth 2Θ, we obtain
sinΘ Θ≅ = cT, Θ= L
cT L
2 2 (4�8)
Impulse Synthetic Aperture Radar 89
Examination of the results in Equations 4�6 and 4�8 suggests that a suitable definition of timed-array 3-dB beamwidth is
Θ =2cT
L (4�9)
which is essentially the same as the expression used in the narrowband case where wave-length λ is substituted by 2 cT—twice the spatial extension of the pulse� In passim, this correspondence 2 cT↔ λ turns out to be valid for most (for instance, see the previously quoted Fraunhofer region definition), if not all parameters describing the performance of pulsed antennas and arrays (Franceschetti, Tatoian, and Gibbs 2005), and was a conjecture advocated over 30 years ago (Franceschetti and Papas 1974)�
Consider the synthesized timed array as depicted in Figure 4�3� The 3-dB beamwidth of the array element is given by 2cT/l, l being its effective length� The array length for the best attainable resolution is equal to the illuminated swath dimension, hence L= (2cT l r), r being the distance of the array from the ground� Accordingly, the azimuth resolution of the timed array is given by
∆x cT
L r cT
cT l rr l
= 2 = 2 =
2 2 2
( ) (4�10a)
where the additional factor 2 in the denominator of the intermediate expression accounts for the round-trip propagation: the time delay Δt between the pulses radiated by nearby elements of the array doubles, virtually reducing the beamwidth of the synthetic array, as shown in Equations 4�5 and 4�6� The final result is identical to that of a conventional SAR�
The range resolution is obviously given by
∆r cT
= 2 (4�10b)
which is the standard expression for a radiated pulse�
All the above derivations, leading to Equations 4�10a and b, are made under the assumption that the imaged point, P, in Figure 4�3 is in the far field, which is not always the case� Accordingly, some processing of the raw data is necessary in order to trans-form the received near-field data to the far-field data; this can be implemented by a
L
P l r 2cT
FIgure 4.3
The synthetic timed array� (From Franceschetti, G�, J� Tatoian, and G� Gibbs, Timed arrays in a nutshell, IEEE Trans Antennas Propagat 53(12):4073–82), © (2005) IEEE�)
90 Advances in Environmental Remote Sensing
simple shift-and-add procedure� If one denotes fn (t), − N ≤ n ≤ N, to be the pulse radiated by the array element n, then the received pulse is
f t f t r na
whereas in the far field it is
f t r n a
the r value being determined by the arrival time of the pulse radiated and received by the array element n = 0� Letting
ρn r na
r r na
= 2+( )2 − 2+( ) 2 (4�13)
and substituting 2ρn/c into Equation 4�11 lead to the conclusion that the azimuthally compressed image of P is given by
g P f t
which justifies naming the procedure shift-and-add� Note that ρn r na
r r na na
= 2+ 2 − 2+ 2 ≅ r2 2
( ) ( ) ( ) (4�15)
if (na/r)2 << 1� This may somehow simplify the procedure, but not significantly�
An example of an experimentally obtained ImpSAR image of an M16 rifle, along with its optical image, is shown in Figure 4�4� In the experiment, the width of the radiated pulse is 100 picoseconds, the length of the timed array is 4�5 m, and the distance between the target and the antenna is 6�0 m�
FIgure 4.4
Optical (top) and microwave (bottom) images of the M16 rifle�
Impulse Synthetic Aperture Radar 91
These parameters imply that range and azimuth resolutions are 1�3 and 8�0 cm, respectively, as shown in Equation 4�10� Note that the image not only resolves fine details of the target, but also has intrinsic peculiarities not present in conventional microwave images� This may be due to the wide bandwidth of the incident signal, so that the illuminated target cannot be modeled simply as a collection of point scatterers� Its resonant response should also be taken into account� This is an open problem worth exploring along two lines: improving the qual-ity of the processed image and extracting value-added information from it (Franceschetti, Tatoian, and Gibbs 2009)�